Vologodsky and Coleman integration on curves with semi-stable reduction
Let X be a curve over a p-adic field K with semi-stable reduction and let ω be a meromorphic differential on X. There are two p-adic integrals one may associate to this data. One is the Vologodsky (abelian, Zarhin, Colmez) integral, which is a global function on the K-points of X defined up to a constant. The other is the collection of Coleman integrals on the subdomains reducing to the various components of the smooth locus. In this talk I will prove the following Theorem, joint with Sarah Zerbes: The Vologodsky integral is given on each subdomain by a Coleman integral, and these integrals are related by the condition that their differences on the connecting annuli form a harmonic 1-cocyle on the edges of the dual graph of the special fiber. I will further explain the implications to the behavior of the Vologodsky integral on the connecting annuli, which has been observed independently and used, by Stoll and Katz-Rabinoff-Zureick-Brown, in works on global bounds on the number of rational points on curves, and an interesting product on 1-forms used in the proof of the Theorem as well as in work on p-adic height pairings. Time permitting I will explain the motivation for this result, which is relevant for the interesting question of generalizing the result to iterated integrals.